Mathematical modeling of the onset of capillary formation initiating angiogenesis. (English) Zbl 0977.92013

Levine, Howard A.; Sleeman, Brian D.; Nilsen-Hamilton, Marit

J. Math. Biol. 42, No. 3, 195-238 (2001).

Summary: It is well accepted that neo-vascular formation can be divided into three main stages (which may be overlapping): (1) changes within the existing vessel, (2) formation of a new channel, (3) maturation of the new vessel. We present a new approach to angiogenesis, based on the theory of reinforced random walks, coupled with a Michaelis-Menten type mechanism which views the endothelial cell receptors as the catalyst for transforming angiogenic factor into proteolytic enzyme in order to model the first stage. In this model, a single layer of endothelial cells is separated by a vascular wall from an extracellular tissue matrix. A coupled system of ordinary and partial differential equations is derived which, in the presence of an angiogenic agent, predicts the aggregation of the endothelial cells and the collapse of the vascular lamina, opening a passage into the extracellular matrix. We refer to this as the onset of vascular sprouting. Some biological evidence for the correctness of our model is indicated by the formation of teats in utero. Further evidence for the correctness of the model is given by its prediction that endothelial cells will line the nascent capillary at the onset of capillary angiogenesis.

MSC:

92C45	Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
60G50	Sums of independent random variables; random walks
35Q92	PDEs in connection with biology, chemistry and other natural sciences
92C37	Cell biology
92C50	Medical applications (general)
92C15	Developmental biology, pattern formation

Keywords:

capillary sprout formation; endothelial cell migration; vascularization; enzyme kinetics; reinforced random walks; Michaelis-Menten